That's fine, but then you shouldn't pretend that you're in a vector space. Words have meanings, and mathematical concepts have precise definitions.

Constraining points to a hypersphere surface only reduces dimensionality by one dimension. For example a sphere in 3-space still has a 2-dimensional surface to play with. That's still a vector space right?

No; it's not closed under linear combinations and it doesn't contain an additive identity.

Well it does if you use the right coordinate system! As the article mentions, most people don't normalize correctly and only get passable answers by luck. You can add locations on the earth's surface if you pick a point to be the origin.